Derive an expression for the radius of `n^(th)` Bohr's orbit of hydrogen atom hence write the expression for the radius of first orbit of hydrogen atom.

Explanation

Consider a atom with effective nuclear charge + Ze. Let an electron revolves around the nucleus with speed v in the orbit of radius r as shown in figure. The necessary centripetal force on electron is provided by the electrostatic force between the electron and the nucleus. Therefore we have, centripetal force= electrostatic force
`(mv^(2))/(r )= (1)/(4pi epsi_(0))(Z e.e)/(r^(2))`
`mv^(2)r= (Ze^(2))/(4pi epsi_(0))` ....(1)
From Bohr.s angular momentum quantization rule,
`mvr= (nh)/(2pi)`
`m^(2)v^(2)r^(2)= (n^(2)h^(2))/(4pi^(2))` ....(2)
Dividing equation (2) by equation (1) we have,
`(m^(2)v^(2)r^(2))/(mv^(2)r)= (n^(2)h^(2))/(4pi^(2)) xx (4pi epsi_(0))/(Ze^(2))`
`mr= (n^(2)h^(2) epsi_(0))/(pi Ze^(2))rArr r= (n^(2)h^(2) epsi_(0))/(pi m Ze^(2))`
For `n^(th)` orbit, `r_(n)= (n^(2)h^(2) epsi_(0))/(pi mZe^(2))`
For H atom Z=1 and for `n^(th)` orbit, `r_(n)= (n^(2)h^(2)epsi_(0))/(pi m e^(2))`